Abstract

The detective quantum efficiency for foveal vision is computed from the flash perception data of Blackwell and McCready. The detective quantum efficiency is identical with the concept of quantum efficiency introduced by Rose in 1946, and is defined as the square of the ratio of the smallest possible threshold to the observed threshold, where the smallest possible threshold is set by the statistical fluctuations in the number of the background photons entering the eye. The computed values of the detective quantum efficiency Q are tabulated in Table V, and depend on the target diameter α, on the light pulse duration T, and on the background luminance B. The maximum values of Q (with the respect to variation of α and T) range from about 0.25% to about 1.0 %over the range from 0.1 to 100 ft-L, with the maximum value occurring at about 1.0 ft-L. The computed values of Q are free of the questionable assumptions previously used by Rose and by Jones regarding integration time and threshold signal-to-noise ratio.

© 1959 Optical Society of America

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References

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  1. R. B. Barnes and M. Czerny, Z. Physik 79, 436–449 (1932).
    [CrossRef]
  2. Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819–840 (1942).
  3. Albert Rose, Proc. Inst. Radio Engrs. 30, 293–300 (1942).
  4. Hessel de Vries, Physica 10, 553–564 (1943).
    [CrossRef]
  5. Albert Rose, J. Soc. Motion Picture Engrs. 47, 273–294 (1946).
  6. Albert Rose, J. Opt. Soc. Am. 38, 196–208 (1948).
    [CrossRef] [PubMed]
  7. Albert Rose, Advances in Electronics 1, 131–166 (1948).
  8. H. B. Barlow, “Retinal noise and absolute threshold,” J. Opt. Soc. Am. 46, 634–639 (1956).
    [CrossRef] [PubMed]
  9. H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).
  10. E. Baumgardt, “Sehmechanismus und Quantenstruktur des Lichtes,” Naturwissenschaften 39, 388–393 (1952).
    [CrossRef]
  11. Hartline, Milne, and Wagman, “Fluctuation of response of single visual sense cells,” Federation Proc. 6, 124 (1947).
  12. M. H. Pirenne, “Physiological mechanisms of vision and the quantum nature of light,” Biol. Revs. Cambridge Phil. Soc. 31, 194–241 (1956).
    [CrossRef]
  13. W. P. Tanner and J. A. Swets, “The human use of information, Part I,” Trans. IRE Prof. Group on Information Theory PGIT-4, 213–221 (1954).
    [CrossRef]
  14. H. R. Blackwell and D. W. McCready (to be published). The writer is indebted to Dr. Blackwell for making these data available prior to publication. Some of these data are published in graphical form by H. R. Blackwell, Illum. Eng.47, 602–609 (1952).
  15. R. Clark Jones, J. Wash. Acad. Sri. 47, 100–108 (1957).
  16. T. C. Fry, Probability and its Engineering Uses (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1929), pp.221–223.
  17. T. G. Birdsall and W. W. Peterson, in Quarterly Progress Report No. 10, University of Michigan, Electronic Defense Group (April, 1954).
  18. H. R. Blackwell, J. Opt. Soc. Am. 36, 624–643 (1946).
    [CrossRef] [PubMed]
  19. Prentice Reeves, J. Opt. Soc. Am. 4, 35–43 (1920).
    [CrossRef]
  20. P. Moon and D. E. Spencer, J. Opt. Soc. Am. 34, 319–329 (1944).
    [CrossRef]
  21. R. Clark Jones, J. Opt. Soc. Am. 37, 879–890 (1947).
  22. R. Clark Jones, J. Opt. Soc. Am. 39, 327–356 (1949).
    [CrossRef]
  23. R. Clark Jones, Advances in Electronics 5, 1–96 (1953).
  24. Albert Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).
    [CrossRef]
  25. David Middleton, Trans Inst. Radio Engrs. IT-3, 86–121 (1957).

1957 (4)

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).

R. Clark Jones, J. Wash. Acad. Sri. 47, 100–108 (1957).

Albert Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).
[CrossRef]

David Middleton, Trans Inst. Radio Engrs. IT-3, 86–121 (1957).

1956 (2)

M. H. Pirenne, “Physiological mechanisms of vision and the quantum nature of light,” Biol. Revs. Cambridge Phil. Soc. 31, 194–241 (1956).
[CrossRef]

H. B. Barlow, “Retinal noise and absolute threshold,” J. Opt. Soc. Am. 46, 634–639 (1956).
[CrossRef] [PubMed]

1954 (1)

W. P. Tanner and J. A. Swets, “The human use of information, Part I,” Trans. IRE Prof. Group on Information Theory PGIT-4, 213–221 (1954).
[CrossRef]

1953 (1)

R. Clark Jones, Advances in Electronics 5, 1–96 (1953).

1952 (1)

E. Baumgardt, “Sehmechanismus und Quantenstruktur des Lichtes,” Naturwissenschaften 39, 388–393 (1952).
[CrossRef]

1949 (1)

1948 (2)

Albert Rose, Advances in Electronics 1, 131–166 (1948).

Albert Rose, J. Opt. Soc. Am. 38, 196–208 (1948).
[CrossRef] [PubMed]

1947 (2)

Hartline, Milne, and Wagman, “Fluctuation of response of single visual sense cells,” Federation Proc. 6, 124 (1947).

R. Clark Jones, J. Opt. Soc. Am. 37, 879–890 (1947).

1946 (2)

H. R. Blackwell, J. Opt. Soc. Am. 36, 624–643 (1946).
[CrossRef] [PubMed]

Albert Rose, J. Soc. Motion Picture Engrs. 47, 273–294 (1946).

1944 (1)

1943 (1)

Hessel de Vries, Physica 10, 553–564 (1943).
[CrossRef]

1942 (2)

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

Albert Rose, Proc. Inst. Radio Engrs. 30, 293–300 (1942).

1932 (1)

R. B. Barnes and M. Czerny, Z. Physik 79, 436–449 (1932).
[CrossRef]

1920 (1)

Barlow, H. B.

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).

H. B. Barlow, “Retinal noise and absolute threshold,” J. Opt. Soc. Am. 46, 634–639 (1956).
[CrossRef] [PubMed]

Barnes, R. B.

R. B. Barnes and M. Czerny, Z. Physik 79, 436–449 (1932).
[CrossRef]

Baumgardt, E.

E. Baumgardt, “Sehmechanismus und Quantenstruktur des Lichtes,” Naturwissenschaften 39, 388–393 (1952).
[CrossRef]

Birdsall, T. G.

T. G. Birdsall and W. W. Peterson, in Quarterly Progress Report No. 10, University of Michigan, Electronic Defense Group (April, 1954).

Blackwell, H. R.

H. R. Blackwell, J. Opt. Soc. Am. 36, 624–643 (1946).
[CrossRef] [PubMed]

H. R. Blackwell and D. W. McCready (to be published). The writer is indebted to Dr. Blackwell for making these data available prior to publication. Some of these data are published in graphical form by H. R. Blackwell, Illum. Eng.47, 602–609 (1952).

H. R. Blackwell and D. W. McCready (to be published). The writer is indebted to Dr. Blackwell for making these data available prior to publication. Some of these data are published in graphical form by H. R. Blackwell, Illum. Eng.47, 602–609 (1952).

Clark Jones, R.

R. Clark Jones, J. Wash. Acad. Sri. 47, 100–108 (1957).

R. Clark Jones, Advances in Electronics 5, 1–96 (1953).

R. Clark Jones, J. Opt. Soc. Am. 39, 327–356 (1949).
[CrossRef]

R. Clark Jones, J. Opt. Soc. Am. 37, 879–890 (1947).

Czerny, M.

R. B. Barnes and M. Czerny, Z. Physik 79, 436–449 (1932).
[CrossRef]

de Vries, Hessel

Hessel de Vries, Physica 10, 553–564 (1943).
[CrossRef]

Fry, T. C.

T. C. Fry, Probability and its Engineering Uses (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1929), pp.221–223.

Hartline,

Hartline, Milne, and Wagman, “Fluctuation of response of single visual sense cells,” Federation Proc. 6, 124 (1947).

Hecht,

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

McCready, D. W.

H. R. Blackwell and D. W. McCready (to be published). The writer is indebted to Dr. Blackwell for making these data available prior to publication. Some of these data are published in graphical form by H. R. Blackwell, Illum. Eng.47, 602–609 (1952).

Middleton, David

David Middleton, Trans Inst. Radio Engrs. IT-3, 86–121 (1957).

Milne,

Hartline, Milne, and Wagman, “Fluctuation of response of single visual sense cells,” Federation Proc. 6, 124 (1947).

Moon, P.

Peterson, W. W.

T. G. Birdsall and W. W. Peterson, in Quarterly Progress Report No. 10, University of Michigan, Electronic Defense Group (April, 1954).

Pirenne,

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

Pirenne, M. H.

M. H. Pirenne, “Physiological mechanisms of vision and the quantum nature of light,” Biol. Revs. Cambridge Phil. Soc. 31, 194–241 (1956).
[CrossRef]

Reeves, Prentice

Rose, Albert

Albert Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).
[CrossRef]

Albert Rose, J. Opt. Soc. Am. 38, 196–208 (1948).
[CrossRef] [PubMed]

Albert Rose, Advances in Electronics 1, 131–166 (1948).

Albert Rose, J. Soc. Motion Picture Engrs. 47, 273–294 (1946).

Albert Rose, Proc. Inst. Radio Engrs. 30, 293–300 (1942).

Shlaer,

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

Spencer, D. E.

Swets, J. A.

W. P. Tanner and J. A. Swets, “The human use of information, Part I,” Trans. IRE Prof. Group on Information Theory PGIT-4, 213–221 (1954).
[CrossRef]

Tanner, W. P.

W. P. Tanner and J. A. Swets, “The human use of information, Part I,” Trans. IRE Prof. Group on Information Theory PGIT-4, 213–221 (1954).
[CrossRef]

Wagman,

Hartline, Milne, and Wagman, “Fluctuation of response of single visual sense cells,” Federation Proc. 6, 124 (1947).

Advances in Biol. and Med. Phys. (1)

Albert Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).
[CrossRef]

Advances in Electronics (2)

Albert Rose, Advances in Electronics 1, 131–166 (1948).

R. Clark Jones, Advances in Electronics 5, 1–96 (1953).

Biol. Revs. Cambridge Phil. Soc. (1)

M. H. Pirenne, “Physiological mechanisms of vision and the quantum nature of light,” Biol. Revs. Cambridge Phil. Soc. 31, 194–241 (1956).
[CrossRef]

Federation Proc. (1)

Hartline, Milne, and Wagman, “Fluctuation of response of single visual sense cells,” Federation Proc. 6, 124 (1947).

J. Gen. Physiol. (1)

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

J. Opt. Soc. Am. (7)

J. Physiol. (London) (1)

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London) 136, 469–488 (1957).

J. Soc. Motion Picture Engrs. (1)

Albert Rose, J. Soc. Motion Picture Engrs. 47, 273–294 (1946).

J. Wash. Acad. Sri. (1)

R. Clark Jones, J. Wash. Acad. Sri. 47, 100–108 (1957).

Naturwissenschaften (1)

E. Baumgardt, “Sehmechanismus und Quantenstruktur des Lichtes,” Naturwissenschaften 39, 388–393 (1952).
[CrossRef]

Physica (1)

Hessel de Vries, Physica 10, 553–564 (1943).
[CrossRef]

Proc. Inst. Radio Engrs. (1)

Albert Rose, Proc. Inst. Radio Engrs. 30, 293–300 (1942).

Trans Inst. Radio Engrs. (1)

David Middleton, Trans Inst. Radio Engrs. IT-3, 86–121 (1957).

Trans. IRE Prof. Group on Information Theory PGIT-4 (1)

W. P. Tanner and J. A. Swets, “The human use of information, Part I,” Trans. IRE Prof. Group on Information Theory PGIT-4, 213–221 (1954).
[CrossRef]

Z. Physik (1)

R. B. Barnes and M. Czerny, Z. Physik 79, 436–449 (1932).
[CrossRef]

Other (3)

T. C. Fry, Probability and its Engineering Uses (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1929), pp.221–223.

T. G. Birdsall and W. W. Peterson, in Quarterly Progress Report No. 10, University of Michigan, Electronic Defense Group (April, 1954).

H. R. Blackwell and D. W. McCready (to be published). The writer is indebted to Dr. Blackwell for making these data available prior to publication. Some of these data are published in graphical form by H. R. Blackwell, Illum. Eng.47, 602–609 (1952).

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Figures (4)

F. 1
F. 1

The Gaussian bell on the left represents the distribution of amplitudes when the signal is absent, and the similar bell on the right is the distribution of signal-plus-noise amplitudes when the signal is present. If the decision threshold is at the position T, the shaded area to the left of the line represents the probability that the device fails to detect a signal that is present, and the shaded area to the right of the line represents the probability that the device judges a signal present when it is not (a false alarm)

F. 2
F. 2

Showing the detective quantum efficiency Q as a function of the target diam α in angular min. The separate curves are for different pulse durations and the label on each curve gives the pulse duration T in sec. All of the curves are for the background luminance of 1.0 ft-L. The top curve is drawn through the highest points of the separate curves of Fig. 3.

F. 3
F. 3

Showing the detective quantum efficiency Q as a function of the light pulse duration T in sec. The separate curves are for different target diam and the label on each curve gives the target diam α in angular min. All of the curves are for a background luminance of 1.0 ft-L. The data plotted in this figure are the same as those plotted in Fig. 2, but with a different abscissa. The top curve is drawn through points that represent the maximum heights of the separate curves in Fig. 2.

F. 4
F. 4

A summary of published data on the detective quantum efficiency of human vision plotted vs the background luminance. Curve A is the result of Rose (1957) and the curves B through D are the results of Jones (1957). Curve E shows the results of this paper. Curves A through D are based mainly on Blackwell’s contrast perception data, and all involve the questionable assumption of an integration period. Curve E is based on the flash perception data of Blackwell and McCready, and is free of this assumption.

Tables (6)

Tables Icon

Table I The entries are the values of the signal-to-noise ratio k required by an ideal detector to achieve a reliability q and a false alarm fraction f.

Tables Icon

Table II The entries are the values of the signal-to-noise ratio k required by an ideal detector to achieve a reliability q in an M-channel forced-choice situation.

Tables Icon

Table III The entries are the values of threshold contract C (for a reliability q = 50% in a four-channel forced-choice situation) with a target diameter α in minutes and a background luminance B in ft-L. All of the entries are for a light pulse duration of 0.1 sec. The values of C greater than 0.5 are in parentheses.

Tables Icon

Table IV Pupil diameters and Stiles-Crawford factors.

Tables Icon

Table V This is the chief table of this report, and contains the main results. The entries in the table show the detective quantum efficiency Q in percent for the target diameter α (in angular min) in the first column, for the target duration T (in seconds) shown at the head of the column, and for the luminance B indicated above that section of the table. The entries that are derived from a contrast C greater than 0.5 are placed in parentheses.

Tables Icon

Table VI The maximum values of the detective quantum efficiency Q for each of four background luminances, and also the target diameters α and the pulse durations T for which the maximum is achieved.

Equations (23)

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C = B s / B .
C = M ¯ s / M ¯ b .
N ( M b M ¯ b ) 2 Av 1 2 = M ¯ b 1 2 ,
S = M ¯ s .
S / N = M ¯ s / M ¯ b 1 2 .
Q = ( S / N ) output 2 M ¯ s 2 / M ¯ b .
Q = k 2 M ¯ b / M ¯ s 2 .
( S / N ) output = F 1 2 M ¯ s / M ¯ b 1 2 ,
Q = F .
y = ( 2 π ) 1 2 exp [ x 2 / 2 ]
y = ( 2 π ) 1 2 exp [ ( x k ) 2 / 2 ] .
q ( p f ) / ( 1 f ) .
y = erf x ( 2 π ) 1 2 x exp ( u 2 / 2 ) d u ,
k = erf 1 ( 1 f ) + erf 1 p
k = erf 1 ( 1 f ) + erf 1 [ f + ( 1 f ) q ] .
M ¯ b = ( π / 16 ) α 2 D 2 S B n T ,
M ¯ s = C M ¯ b ,
Q = 16 k 2 / π α 2 C 2 D 2 S B n T .
Q = 5.59 × 10 12 k 2 / α 2 C 2 D 2 S B n T .
k = 1.22 .
n = 4.073 × 10 15 photons per lu‐sec .
Q = 0.0020434 / α 2 C 2 D 2 S B T .
Q = 0.007441 = 0.7441 % .